Open Access
Access granted
Subscription Access
Vol 80, No 2 (2025)
- Year: 2025
- Articles: 9
- URL: https://journals.rcsi.science/0042-1316/issue/view/20353
Differential geometry on finitely presented groups and related combinatorial invariants
Abstract
Many concepts in differential geometry (for example, Riemannian metric, length, shortest arc, volume), defined originally for manifolds, are easily transferred to simplicial polyhedra. This extends significantly the scope of application of such concepts familiar in the framework of manifolds as systolic volume and volume entropy. Many properties of these invariants depend only on the fundamental group of the polyhedron. Using the process of minimisation, it becomes possible to transfer these invariants directly to finitely presented groups. In studying the systolic area and volume entropy of groups, new combinatorial invariants arise, which are of independent interest. The paper is an introduction to this area at the intersection of geometry, topology, and group theory.Bibliography: 65 titles.
Uspekhi Matematicheskikh Nauk. 2025;80(2):3-50
3-50
51-122
Special Bohr–Sommerfeld geomery
Abstract
This survey sums up a cycle of papers devoted to the construction of finite-dimensional moduli spaces points in which are certain special Lagrangian submanifolds of compact complex simply connected algebraic varieties. The starting point for this construction was the idea, due to A. Tyurin, to treat Largrangian submanifolds (or equivalence classes of such submanifolds) as mirror counterparts of stable vector bundles. Our constructions are based on the programme of abelian Lagrangian algebraic geometry developed by A. Tyurin and Gorodentsev 25 years ago. Since this programme was in its turn based on the Bohr–Sommerfeld Lagrangian geometry known in geometric quantization, we call our construction special Bohr–Sommerfeld geometry. The definitions arising in the course of work turn out to be closely connected with the theory of Weinstein domains, Eliashberg's conjectures, and many other concepts in symplectic geometry. The core conjecture that arose in our work and is confirmed by the available examples states that each moduli space of this type is in its turn an algebraic variety.Bibliography: 13 titles.
Uspekhi Matematicheskikh Nauk. 2025;80(2):123-164
123-164
On the intersection of finitely generated varieties of monoids
Uspekhi Matematicheskikh Nauk. 2025;80(2):165-166
165-166
Universal matrix Capelli identity
Uspekhi Matematicheskikh Nauk. 2025;80(2):167-168
167-168
Asymptotic properties of polynomials defined by shifted orthogonality conditions
Uspekhi Matematicheskikh Nauk. 2025;80(2):169-170
169-170
On the 80th birthday of Yulij Sergeevich Ilyashenko
Uspekhi Matematicheskikh Nauk. 2025;80(2):171-183
171-183
Valerii Vladimirovich Volchkov (on his sixtieth birthday)
Uspekhi Matematicheskikh Nauk. 2025;80(2):184-189
184-189
International (56th National) School and Conference “Modern problems in mathematics and its applications”
Uspekhi Matematicheskikh Nauk. 2025;80(2):190-190
190-190